# The energy associated with electric field

Question:

The energy associated with electric field is $\left(U_{E}\right)$ and with magnetic fields is $\left(U_{B}\right)$ for an electromagnetic wave in free space. Then :

1. (1) $\quad U_{E}=\frac{U_{B}}{2}$

2. (2) $\mathrm{U}_{\mathrm{E}}>\mathrm{U}_{\mathrm{B}}$

3. (3) $\mathrm{U}_{\mathrm{E}}<\mathrm{U}_{\mathrm{B}}$

4. (4) $\mathrm{U}_{\mathrm{E}}=\mathrm{U}_{\mathrm{B}}$

Correct Option: , 4

Solution:

(4) Average energy density of magnetic field,

$u_{B}=\frac{B_{0}^{2}}{4 \mu_{0}}$

Average energy density of electric field,

$\mathrm{u}_{\mathrm{E}}=\frac{\varepsilon_{0} \mathrm{E}_{0}^{2}}{4}$

Now, $\mathrm{E}_{0}=\mathrm{CB}_{0}$ and $\mathrm{C}^{2}=\frac{1}{\mu_{0} \in_{0}}$

$\mathrm{u}_{\mathrm{E}}=\frac{\varepsilon_{0}}{4} \times \mathrm{C}^{2} \mathrm{~B}_{0}^{2}=\frac{\varepsilon_{0}}{4} \times \frac{1}{\mu_{0} \varepsilon_{0}} \times \mathrm{B}_{0}^{2}=\frac{\mathrm{B}_{0}^{2}}{4 \mu_{0}}=\mathrm{u}_{\mathrm{B}}$

$\therefore \mathrm{u}_{\mathrm{E}}=\mathrm{u}_{\mathrm{B}}$

Since energy density of electric and magnetic field is same, so energy associated with equal volume will be equal i.e. $u_{E}=u_{B}$